Notice that the two results issued from [1]} in the latter proof still hold true if \(\partial K_\rho \cap \partial \Omega \ne \emptyset \) . The interested reader may check easily that all the other computations remain true replacing \(K_\rho \)  by \(\Omega \) .
In this case, (REF ) reads
\(\int _{\Omega } \vert \nabla u_i \vert ^2 \le C \bigl (\delta _i- \frac{\ell {K_{i,-i}^+}^2}{4K_{i,i}^-} \bigr )^{-1}\Vert Q \Vert _{L^{s/(s-1)}(\Omega )}\) 
